74 2. Elliptic curves
Exercise 2.12.19. Let e1,e2,e3 be three distinct integers. Show that
Δ = (e1 − e2)(e2 − e3)(e1 − e3) is always even.
Exercise 2.12.20. In this exercise we study the structure of the
quotient G/2G, where G is a finite abelian group.
(1) Let p ≥ 2 be a prime and let G =
Z/peZ,
with e ≥ 1. Prove
that G/2G is trivial if and only if p 2.
(2) Prove that, if G =
Z/2eZ
and e ≥ 1, then G/2G
∼
= Z/2Z.
(3) Finally, let G be an arbitrary finite abelian group. We define
G[2∞] to be the 2-primary component of G, i.e.,
G[2∞]
= {g ∈ G :
2n
· g = 0 for some n ≥ 1}.
In other words, G[2∞] is the subgroup of G formed by those
elements of G whose order is a power of 2. Prove that
G[2∞]
∼
=
Z/2e1
Z ⊕
Z/2e2
Z ⊕ · · · ⊕
Z/2er
Z
for some r ≥ 0 and ei ≥ 1 (here r = 0 means G[2∞] is
trivial). Also show that G/2G
∼
=
(Z/2Z)r.
Exercise 2.12.21. Show that the space
C :
2Y 2 − X2 = 34,
Y
2
−
Z2
= 34
does not have any rational solutions with X, Y, Z ∈ Q. (Hint: modify
the system so there are no powers of 2 in any of the denominators,
then work modulo 8.)
Exercise 2.12.22. For the following elliptic curves, use the method
of 2-descent (as in Proposition 2.10.3 and Example 2.10.4) to find the
rank of E/Q and generators of E(Q)/2E(Q). Do not use Sage:
(1) E :
y2
=
x3
− 14931x + 220590.
(2) E :
y2
=
x3
−
x2
− 6x.
(3) E :
y2
=
x3
− 37636x.
(4) E :
y2
=
x3
−
962x2
+ 148417x. (Hint: use Theorem 2.7.4
first to find a bound on the rank.)
Exercise 2.12.23. Find the rank and generators for the rational
points on the elliptic curve
y2
= x(x + 5)(x + 10).
